This paper presents grey wolf optimization (GWO) to solve convex economic load dispatch (ELD) problem. GreyWolf Optimization (GWO) is a new meta-heuristic inspired by grey wolves. The leadership hierarchy and huntingmechanism of the grey wolves is mimicked in GWO. The objective of ELD problem is to minimize the totalgeneration cost while fulfilling the different constraints, when the required load of power system is beingsupplied. The proposed technique is implemented on two different test systems for solving the ELD with variousload demands. To show the effectiveness of GWO to solve ELD problem results were compared with otherexisting techniques.
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Dr.Sudhir Sharma et al. Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp.128-132
RESEARCH ARTICLE
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OPEN ACCESS
Economic Load Dispatch Using Grey Wolf Optimization
Dr.Sudhir Sharma1,Shivani Mehta2, Nitish Chopra3
Associate Professor1, Assistant Professor2, Student, Master of Technology3 Department of Electrical
Engineering, D.A.V.I.E.T., Jalandhar, Punjab, India 1, 2, 3
ABSTRACT
This paper presents grey wolf optimization (GWO) to solve convex economic load dispatch (ELD) problem. Grey
Wolf Optimization (GWO) is a new meta-heuristic inspired by grey wolves. The leadership hierarchy and hunting
mechanism of the grey wolves is mimicked in GWO. The objective of ELD problem is to minimize the total
generation cost while fulfilling the different constraints, when the required load of power system is being
supplied. The proposed technique is implemented on two different test systems for solving the ELD with various
load demands. To show the effectiveness of GWO to solve ELD problem results were compared with other
existing techniques.
Keywords: economic load dispacth;GWO; transmission loss
I. INTRODUCTION
Electrical power plays a pivotal role in the
modern world to satisfy various needs. It is therefore
very important that the electrical power generated is
transmitted and distributed efficiently in order to
satisfy the power requirement. Electrical power is
generated in several ways. The economic scheduling
of all generators in a system to meet desired demand
is important problem in operation and planning of
power system. The Economic Load Dispatch (ELD)
problem is the most significant optimization problem
in scheduling the generation of thermal generators in
power system. In ELD problem, ultimate goal is to
decrease the operation cost of the power generation
system, while supplying the required power
demanded. In addition to this, the various operational
constraints of the system should also be satisfied.
Traditional methods to solve ELD problem include
the linear programming method, gradient method,
lambda iteration method and Newton‟s method [1].
Dynamic programming is one of the techniques
to solve ELD problem, but it suffer from problem of
irritation of dimensionality [2]. Meta-heuristic
techniques, such as genetic algorithms [3-5],
differential evolution [6] , tabu search [7] ,simulated
annealing [8], particle swarm optimization (PSO) [9],
biogeography-based optimization [10],intelligent
water drop algorithm[11] ,harmony search[12]
,gravitational
search
algorithm[13],firefly
algorithm[14],hybrid gravitational search[15],cuckoo
search (CS) [16],modified harmony search[17] have
been successfully applied to ELD problems. Recently,
a new meta-heuristic technique called grey wolf
optimization has been proposed by Mirjalili et al.,
[18]. In this paper the ELD problem has been solved
by using grey wolf optimization.
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II. PROBLEM FORMULATION
The objective function of the ELD problem is to
minimize the total generation cost while satisfying the
different constraints, when the required load of power
system is being supplied. The objective function to be
minimized is given by the following equation:
n
F ( Pg ) (ai Pgi2 bi Pgi ci )
… (1)
i 1
The overall fuel cost has to be reduced with the
following constraints:
1) Power balance constraint
The total generation by all the generators must be
equal to the total power demand and system‟s real
power loss.
𝑛
….. (2)
𝑖=1 𝑃𝑔𝑖 − 𝑃𝑑 − 𝑃𝑙
2) Generator limit constraint
The real power generation of each generator is to be
controlled inside its particular upper and lower
operating limits.
𝑃𝑔𝑖𝑚𝑖𝑛 ≤ 𝑃𝑔𝑖 ≤ 𝑃𝑔𝑖𝑚𝑎𝑥 i=1,2,...,ng
…..(3)
Where
ai, bi, ci : coefficient of fuel cost of ith generator,
Rs/MW2 h, Rs/MW h, Rs/h
F(Pg ) : total fuel cost, Rs/h
n
: number of generators
𝑚𝑖𝑛
𝑃𝑔𝑖 : Minimum limit of generation for ith generator,
MW
𝑃𝑔𝑖𝑚𝑎𝑥 : Maximum limit of generation for ith generator,
MW
Pl
: Transmission losses, MW
Pd
: Power demand, MW
III. Grey Wolf Optimization (GWO)
The GWO is firstly proposed by Mirjalili et al.,
[18]. The algorithm was inspired by the democratic
behavior and the hunting mechanism of grey wolves
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Dr.Sudhir Sharma et al. Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp.128-132
in the wild. In a pack, the grey wolves follow very
firm social leadership hierarchy. The leaders of the
pack are a male and female, are called alpha (α). The
second level of grey wolves, which are subordinate
wolves that help the leaders, are called beta (β).
Deltas (δ) are the third level of grey wolves which
has to submit to alphas and betas, but dominate the
omega. The lowest rank of the grey wolf is omega
(ω), which have to surrender to all the other
governing wolves. The GWO algorithm is provided
in the mathematical models as follows:
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4) Search for prey and attacking prey
The „A‟ is an arbitrary value in the gap [-2a, 2a].
When |A| < 1, the wolves are forced to attack the
prey. Attacking the prey is the exploitation ability
and searching for prey is the exploration ability. The
random values of „A‟ are utilized to force the search
agent to move away from the prey. When |A| > 1, the
grey wolves are enforced to diverge from the prey.
1) Social hierarchy
In the mathematical model of the social hierarchy of
the grey wolves, alpha (α) is considered as the fittest
solution. Accordingly, the second best solution is
named beta (β) and third best solution is named delta
(δ) respectively. The candidate solutions which are
left over are taken as omega (ω). In the GWO, the
optimization (hunting) is guided by alpha, beta, and
delta. The omega wolves have to follow these
wolves.
2) Encircling prey
The grey wolves encircle prey during the hunt.
The encircling behavior can be mathematically
modeled as follows [18]:
D = C. Xp t − X(t)
… (4)
X t + 1 = X p t − A. D
… (5)
Where A and C are coefficient vectors, X p is the
prey‟s position vector, X denotes the grey wolf‟s
position vector and „t‟ is the current iteration.
The calculation of vectors A and C is done as follows
[18]:
A = 2. a. r1 . a
… (6)
C = 2. r2
… (7)
Where values of „a ‟are linearly reduced from 2 to 0
during the course of iterations and r1, r2 are arbitrary
vectors in gap [0, 1].
3) Hunting
The hunt is usually guided by the alpha, beta and
delta, which have better knowledge about the
potential location of prey. The other search agents
must update their positions according to best search
agent‟s position. The update of their agent position
can be formulated as follows [18]:
Dα = C1 . X α − X
Dβ = C 2 . X β − X
… (8)
Dδ = C 3 . X δ − X
X1 = Xα − A1 . (Dα )
X 2 = X β − A 2 . Dβ
… (9)
X3 = Xδ − A3 . (Dδ )
X t+1 =
X 1 +X 2 +X 3
3
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… (10)
Fig 3.1: flowchart of GWO
IV. RESULTS & DISCUSSIONS
GWO has been used to solve the ELD problems
in two diverse test cases for exploring its
optimization potential, where the objective function
was limited within power ranges of the generating
units and transmission losses were also taken into
account. The iterations performed for each test case
are 500 and number of search agents (population)
taken in both test cases is 30.
1) Test system I: Three generating units
The input data for three generators and loss
coefficient matrix Bmn is derived from reference [16]
and is given in table 4.1.The economic load dispatch
for 3 generators is solved with GWO and results are
compared with lambda iteration and cuckoo search.
129 | P a g e
Dr.Sudhir Sharma et al. Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp.128-132
0.000071
Bmn= 0.000030
0.000025
Table 4.1: Generating unit data for test case I
Uni
ai
bi
ci
𝑷𝒎𝒂𝒙
𝑷𝒎𝒊𝒏
𝒈𝒊
𝒈𝒊
t
1
0.0354 38.3055 1243.531
35
210
6
3
1
2
0.0211 36.3278 1658.569 130
325
1
2
6
3
0.0179 38.2704 1356.659 125
315
9
1
2
0.000030
0.000069
0.000032
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0.000025
0.000032
0.000080
Table 4.2: GWO results for 3-unit system
Sr.no.
Techniques
1
CS[16]
GWO
2
Power
demand
(MW)
350
CS[16]
GWO
3
450
CS[16]
GWO
500
P1(MW)
P2(MW)
P3(MW)
PLoss (MW)
70.3012
156.267
129.208
5.77698
Fuel Cost
(Rs/hr)
18564.5
70.30259
156.289
129.184
5.77696
18564.483
93.9374
193.814
171.862
9.6127
23112.4
93.9362
193.8043
171.872
9.6127
23112.363
105.88
212.728
193.306
11.9144
25465.5
105.8848
212.7137
193.3157
11.91434
25465.469
Table 4.3: Comparison results of GWO for 3-Unit system
Fuel Cost (Rs/hr)
Power demand
Lambda Iteration
Cuckoo Search
(MW)
Method [16]
Algorithm [16]
350
18570.7
18564.5
Sr.no.
1
2
3
450
500
23146.8
25495.2
2) Test system II: Six generating units
The input data for six generators and loss
coefficient matrix Bmn is derived from reference
[16] and is given in table 4.4.The economic load
dispatch for 6 generators is solved with GWO and
results are compared with conventional quadratic
programming, lambda iteration, particle swarm
optimization and cuckoo search.
Table 4.4: Generating unit data for test case II
bi
ci
𝑷𝒎𝒊𝒏
𝒈𝒊
𝑷𝒎𝒂𝒙
𝒈𝒊
125
150
225
210
325
315
0.000022
0.000020
0.000019
0.000025
0.000032
0.000085
130 | P a g e
Dr.Sudhir Sharma et al. Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp.128-132
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Table 4.5: GWO results for 6-unit system
Sr.no
.
Techni
ques
1
Convent
ional[9]
CS[16]
2
3
Power
Dema
nd
(MW)
P1
(MW)
P2
(MW
)
P3
(MW)
P4
(MW)
P5
(MW)
P6
(MW)
PLoss
(MW)
Fuel
Cost
(Rs/hr)
23.90
10.00
95.63
100.70
202.82
182.02
15.07
32096.58
23.8603
10
95.6389
100.708
202.832
181.198
14.2374
32094.7
GWO
23.911
10
95.571
100.740
202.752
181.261
14.2373
32094.67
Convent
ional[9]
CS[16]
28.33
10.00
118.95
118.67
230.75
212.80
19.50
36914.01
28.2908
10.00
118.958
118.675
230.763
212.745
19.4319
36912.2
GWO
28.3514
10.00
118.887
118.748
230.704
212.737
19.4308
36912.14
Convent
ional[9]
CS[16]
32.63
14.48
141.54
136.04
257.65
243.00
25.34
41898.45
32.5861
14.48
43
14.86
60
141.548
136.045
257.664
243.009
25.3309
41896.7
141.502
1
136.0254
257.518
242.8632
25.3165
41896.63
600
700
800
GWO
Sr.no.
1
32.5408
Table 4.6: Comparison results of GWO for 6-Unit system
Fuel Cost (Rs/hr)
Power
Lambda
Conventional
PSO[9]
Cuckoo
demand
Iteration
Method[9]
Search
(MW)
Method[16]
Algorithm[16]
600
32129.8
32096.58
32094.69
32094.7
Grey Wolf
Optimization
32094.67
2
700
36946.4
36914.01
36912.16
36912.2
36912.145
3
800
41959.0
41898.45
41896.66
41896.9
41896.632
Fig 4.1: Convergence characteristics of test system I
with 500MW demand
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Fig 4.2: Convergence characteristics of test
system II with 800MW demand
131 | P a g e
Dr.Sudhir Sharma et al. Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp.128-132
V. CONCLUSION
In this paper economic load dispatch problem
has been solved by using GWO. The results of GWO
are compared for three and six generating unit
systems with other techniques. The algorithm is
programmed in MATLAB(R2009b) software
package. The results show effectiveness of GWO for
solving the economic load dispatch problem. The
advantage of GWO algorithm is its simplicity,
reliability and efficiency for practical applications.
[11]
[12]
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